Entanglement scaling of excited states in large one-dimensional many-body localized systems

We study the properties of excited states in one-dimensional many-body localized (MBL) systems using a matrix product state algorithm. First, the method is tested for a large disordered noninteracting system, where for comparison we compute a quasiexact reference solution via a Monte Carlo sampling of the single-particle levels. Thereafter, we present extensive data obtained for large interacting systems of $L\sim 100$ sites and large bond dimensions $\chi \sim 1700$, which allows us to quantitatively analyze the scaling behavior of the entanglement $S$ in the system. The MBL phase is characterized by a logarithmic growth $S\left(L\right)\sim log\left(L\right)$ over a large scale separating the regimes where volume and area laws hold. We check the validity of the eigenstate thermalization hypothesis. Our results are consistent with the existence of a mobility edge.

Figure 1

Convergence tests for the DMRG algorithm in the MBL phase. (a) Magnetization at the center site as a function of the DMRG sweeps (the magnetization reached at the end of the simulation is subtracted). (b) Mutual overlap of the states obtained after two consecutive sweeps.

Figure 2

Proof-of-principle application of the excited-state DMRG algorithm to a noninteracting system $\left(\mathrm{\Delta }=0\right)$ with disorder $\eta =5$. (a) System-size dependence of the entanglement entropy in highly excited states with an energy density $E/L\approx -0.125$; the area law holds. Four different disorder realizations are shown at each $L$ [$4\times$(number of $L$ points) realizations in total]. The data was obtained using a mixing $\lambda$ strictly chosen according to the simple form $E=\lambda /\left[2\right(\lambda -1\left)\right]$. (b) The same but for a constant system size $L=200$ and four different disorder configurations.

Figure 3

Scatter plot of the bond dimensions $\chi$ occurring in the DMRG calculation of a highly excited state with energy $E/L=-0.125$ in a noninteracting system $\left(\mathrm{\Delta }=0\right)$ of size $L$. Panels (a)–(d) show various disorder strengths $\eta$; the total number of configurations in each panel is $O\left(5000\right)$.

Figure 4

Scaling of the bipartite spin fluctuations $F\left(L\right)$ corresponding to the data sets shown in Fig. 3. For comparison, an analytic reference solution is constructed from the exact diagonalization of the noninteracting Hamiltonian combined with a Monte Carlo sampling of the single-particle levels to determine excited states with $E/L=-0.125$. The error bars are defined via the standard deviation of $F$.

Figure 5

Scaling of the entanglement entropy in a highly excited state with $E/L=-0.125$. (a) Exact result for the noninteracting system $\left(\mathrm{\Delta }=0\right)$ and various $\eta$. Upon switching on the disorder, the volume law $S\left(L\right)\sim L$ is replaced by a logarithmic increase $S\left(L\right)\sim log\left(L\right)$ (see the inset). The area law $S\left(L\right)\sim const$ only manifests on a much larger scale [see Fig. 6]. The error is of the order of the linewidth. (b) Comparison of $\mathrm{\Delta }=0$ with the interacting case $\mathrm{\Delta }=0.5$ at fixed $\eta =5$. In the former case, both the exact solution and DMRG data are shown.

Figure 6

(a) Data from Fig. 5 for $\eta =1.25$ on a magnified scale. (b) Growth of the average bond dimension during a DMRG calculation for the same parameters ($\sim 100$ samples at each $L$). The maximally allowed bond dimension is $\chi \sim 1700$.

Figure 7

Same as in Fig. 3 but for an isotropic $XXZ$ chain $\left(\mathrm{\Delta }=1\right)$, fixed disorder $\eta =2.5$, but various energies $E/L$. Each panel shows data for $O\left(5000\right)$ configurations.

Figure 8

Scaling of the average bond dimension. The metallic phase is characterized by $\chi \sim e^L$; the entanglement features a volume law $S\left(L\right)\sim L$. In the many-body localized regime, our data suggests $\chi \sim L$ and hence a logarithmic scaling of the entanglement on this length scale. (a) Fixed, intermediate-strength disorder $\eta =2.5$ and various excited-state energies $E/L$ for an isotropic $XXZ$ chain $(\mathrm{\Delta }=1$). The inset shows the same data for the largest energies on a log-linear scale. The results are consistent with the existence of a mobility edge. (b),(c) Fixed $E/L=-0.025$, various $\mathrm{\Delta }$, and larger values of $\eta =5$ and $\eta =7$ (inset) deeper in the MBL phase. The system is localized.

Figure 9

Probability distributions $P\left(S\right)$ and $P\left(F\right)$ for the entanglement and the bipartite spin fluctuations of a system in the MBL phase with $\eta =7,\mathrm{\Delta }=1,E/L=-0.025$, and $L=10$ (solid) or $L=20$ (dashed). The data was obtained using $O\left(3000\right)$ samples.

Figure 10

Probing the eigenstate thermalization hypothesis via the expectation value of the local magnetization $\langle S_l^z\rangle$ for $\mathrm{\Delta }=1$ and at various energies. (a) Single configuration drawn at $\eta =3$ for a chain of $L=20$ sites (five different positions $l\approx L/2$ are shown; the disorder configuration is the same at each $E$). One observes a crossover between thermal and nonergodic behavior as the energy decreases. (b) Two single configurations drawn at $\eta =10$ for a large chain with $L=100$ (only one position $l=L/2$ is shown for each). States at all energies are localized and nonergodic.